Infinitary Logic and Inductive Definability Over Finite Structures

Infinitary Logic and Inductive Definability Over Finite Structures

<p>University of Pennsylvania </p><p><a href="/goto?url=https://repository.upenn.edu/" target="_blank">ScholarlyCommons </a></p><p><a href="/goto?url=https://repository.upenn.edu/cis_reports" target="_blank">Technical Reports (CIS) </a></p><p>November 1991 </p><p><a href="/goto?url=https://repository.upenn.edu/cis" target="_blank">Department of Computer &amp; Information Science </a></p><p>Infinitary Logic and Inductive Definability Over Finite Structures </p><p>Anuj Dawar </p><p>University of Pennsylvania </p><p>Steven Lindell </p><p>University of Pennsylvania </p><p>Scott Weinstein </p><p>University of Pennsylvania </p><p>Follow this and additional works at: <a href="/goto?url=https://repository.upenn.edu/cis_reports?utm_source=repository.upenn.edu%2Fcis_reports%2F365&amp;utm_medium=PDF&amp;utm_campaign=PDFCoverPages" target="_blank">https:</a><a href="/goto?url=https://repository.upenn.edu/cis_reports?utm_source=repository.upenn.edu%2Fcis_reports%2F365&amp;utm_medium=PDF&amp;utm_campaign=PDFCoverPages" target="_blank">/</a><a href="/goto?url=https://repository.upenn.edu/cis_reports?utm_source=repository.upenn.edu%2Fcis_reports%2F365&amp;utm_medium=PDF&amp;utm_campaign=PDFCoverPages" target="_blank">/</a><a href="/goto?url=https://repository.upenn.edu/cis_reports?utm_source=repository.upenn.edu%2Fcis_reports%2F365&amp;utm_medium=PDF&amp;utm_campaign=PDFCoverPages" target="_blank">repository.upenn.edu/cis_reports </a></p><p>Recommended Citation </p><p>Anuj Dawar, Steven Lindell, and Scott Weinstein, "Infinitary Logic and Inductive Definability Over Finite Structures", . November 1991. </p><p>University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-91-97. </p><p>This paper is posted at ScholarlyCommons. <a href="/goto?url=https://repository.upenn.edu/cis_reports/365" target="_blank">https:</a><a href="/goto?url=https://repository.upenn.edu/cis_reports/365" target="_blank">/</a><a href="/goto?url=https://repository.upenn.edu/cis_reports/365" target="_blank">/</a><a href="/goto?url=https://repository.upenn.edu/cis_reports/365" target="_blank">repository.upenn.edu/cis_reports/365 </a></p><p>For more information, please contact <a href="mailto:[email protected]" target="_blank">[email protected]</a>. </p><p>Infinitary Logic and Inductive Definability Over Finite Structures </p><p>Abstract </p><p>The extensions of first-order logic with a least fixed point operators (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu [AV91b] investigated the relation of these two logics in the absence of an ordering, using a mchine model of generic computation. In particular, they showed that the two languages have equivalent expressive power if and only if P = PSPACE. These languages can also be seen as fragments of an infinitary logic where each </p><p>ω</p><p></p><ul style="display: flex;"><li style="flex:1">formula has a bounded number of variables, L </li><li style="flex:1">(see, for instance, [KV90]). We present a treatment of </li></ul><p></p><p>∞ω </p><p>the results in [AV91b] from this point of view. In particular, we show that we can write a formula of FO + LFP and P from ordered structures to classes of structures where every element is definable. We also settle a conjecture mentioned in [AV91b] by showing that FO + LFP in properly contained in the </p><p>ω</p><p>polynomial time computable fragment of L recursively enumerable class. <br>, raising the question of whether the latter fragment is a </p><p>∞ω </p><p>Comments </p><p>University of Pennsylvania Department of Computer and Information Science Technical Report No. MS- CIS-91-97. </p><p>This technical report is available at ScholarlyCommons: <a href="/goto?url=https://repository.upenn.edu/cis_reports/365" target="_blank">https:</a><a href="/goto?url=https://repository.upenn.edu/cis_reports/365" target="_blank">/</a><a href="/goto?url=https://repository.upenn.edu/cis_reports/365" target="_blank">/</a><a href="/goto?url=https://repository.upenn.edu/cis_reports/365" target="_blank">repository.upenn.edu/cis_reports/365 </a></p>

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